We now discuss two related notions:

- The flow of something across the boundary of the region.
- The change in the amount of something inside the region.

When we consider a conserved quantity such as energy, momentum, or charge, these two notions stand in a one-to-one relationship. In general, though, these two notions are not equivalent.

In particular, consider equation 6.32, which is restated here:

dE = −P dV + T dS + advection
(15.1) |

Although officially **d***E* represents the change in energy in the
interior of the region, we are free to interpret it as the flow of
energy across the boundary. This works because *E* is a conserved
quantity.

The advection term is explicitly a boundary-flow term.

It is extremely tempting to interpret the two remaining terms as boundary-flow terms also … but this is not correct!

Officially *P***d***V* describes a property of the interior of the
region. Ditto for *T***d***S*. Neither of these can be converted to a
boundary-flow notion, because neither of them represents a conserved
quantity. In particular, *P***d***V* energy can turn into *T***d***S*
energy entirely within the interior of the region, without any
boundary being involved.

Let’s be clear: boundary-flow ideas are elegant, powerful, and widely
useful. Please don’t think I am saying anything bad about
boundary-flow ideas. I am just saying that the *P***d***V* and *T***d***S*
terms *do not represent* flows across a boundary.

Misinterpreting *T***d***S* as a boundary term is a ghastly mistake. It
is more-or-less tantamount to assuming that heat is a conserved
quantity unto itself. It would set science back over 200 years,
back to the “caloric” theory.

Once these mistakes have been pointed out, they seem obvious, easy to spot, and easy to avoid. But beware: mistakes of this type are extremely prevalent in introductory-level thermodynamics books.